On ABS Estrada index of trees
Let G be a graph with n vertices, and d i be the degree of its i -th vertex. The ABS matrix of G is the square matrix of order n whose ( i , j )-entry is equal to ( d i + d j - 2 ) / ( d i + d j ) if the i -th vertex and the j -th vertex of G are adjacent, and 0 otherwise. Let ρ 1 ≥ ρ 2 ≥ ⋯ ≥ ρ n b...
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| Veröffentlicht in: | Journal of applied mathematics & computing 2024-12, Vol.70 (6), p.5483-5495 |
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| container_title | Journal of applied mathematics & computing |
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| creator | Lin, Zhen Zhou, Ting Liu, Yingke |
| description | Let
G
be a graph with
n
vertices, and
d
i
be the degree of its
i
-th vertex. The
ABS
matrix of
G
is the square matrix of order
n
whose (
i
,
j
)-entry is equal to
(
d
i
+
d
j
-
2
)
/
(
d
i
+
d
j
)
if the
i
-th vertex and the
j
-th vertex of
G
are adjacent, and 0 otherwise. Let
ρ
1
≥
ρ
2
≥
⋯
≥
ρ
n
be the eigenvalues of the
ABS
matrix of
G
. Then the
ABS
Estrada index of
G
, denoted by
E
ABS
(
G
)
, is defined as
E
ABS
(
G
)
=
∑
i
=
1
n
e
ρ
i
. In this paper, the chemical importance of the
ABS
Estrada index is investigated and it is shown that the predictive ability of
ABS
Estrada index is stronger than other connectivity Estrada indices (including Randić Estrada index, harmonic Estrada index and
ABC
Estrada index) and
ABS
index for octane isomers. Since chemical graphs of octane isomers are trees, we study the extremal problem of
ABS
Estrada index of trees, and prove that for any tree
T
n
with
n
≥
3
vertices,
E
ABS
(
P
n
)
≤
E
ABS
(
T
n
)
≤
E
ABS
(
K
1
,
n
-
1
)
with equality in the left (resp., right) inequality if and only if
T
n
is isomorphic to the path
P
n
(resp., the star
K
1
,
n
-
1
). |
| doi_str_mv | 10.1007/s12190-024-02188-z |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_3134326323</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>3134326323</sourcerecordid><originalsourceid>FETCH-LOGICAL-c200t-86dc4da5a4a309779cd9a0f8c9ae30da4a44b16910f59fcb67c1003a1fe12eb03</originalsourceid><addsrcrecordid>eNp9kFtLAzEQhYMoWKt_QBAWfI7O5LKbPNZSL1Dog_ocsrlIi-7WZAvaX290Bd98GGY4nDMzfIScI1whQHOdkaEGCkyUQqXo_oBMUNWSMlDysMxSKyqLcExOct4A1I0GPSEXq66a3TxWizwk62217nz4qPpYDSmEfEqOon3N4ey3T8nz7eJpfk-Xq7uH-WxJHQMYqKq9E95KKywH3TTaeW0hKqdt4OCLLESLtUaIUkfX1o0rX3OLMSALLfApuRz3blP_vgt5MJt-l7py0nDkgrOaM15cbHS51OecQjTbtH6z6dMgmG8MZsRgCgbzg8HsS4iPoVzM3UtIf6v_SX0Bn8peDg</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>3134326323</pqid></control><display><type>article</type><title>On ABS Estrada index of trees</title><source>Springer Nature - Complete Springer Journals</source><creator>Lin, Zhen ; Zhou, Ting ; Liu, Yingke</creator><creatorcontrib>Lin, Zhen ; Zhou, Ting ; Liu, Yingke</creatorcontrib><description>Let
G
be a graph with
n
vertices, and
d
i
be the degree of its
i
-th vertex. The
ABS
matrix of
G
is the square matrix of order
n
whose (
i
,
j
)-entry is equal to
(
d
i
+
d
j
-
2
)
/
(
d
i
+
d
j
)
if the
i
-th vertex and the
j
-th vertex of
G
are adjacent, and 0 otherwise. Let
ρ
1
≥
ρ
2
≥
⋯
≥
ρ
n
be the eigenvalues of the
ABS
matrix of
G
. Then the
ABS
Estrada index of
G
, denoted by
E
ABS
(
G
)
, is defined as
E
ABS
(
G
)
=
∑
i
=
1
n
e
ρ
i
. In this paper, the chemical importance of the
ABS
Estrada index is investigated and it is shown that the predictive ability of
ABS
Estrada index is stronger than other connectivity Estrada indices (including Randić Estrada index, harmonic Estrada index and
ABC
Estrada index) and
ABS
index for octane isomers. Since chemical graphs of octane isomers are trees, we study the extremal problem of
ABS
Estrada index of trees, and prove that for any tree
T
n
with
n
≥
3
vertices,
E
ABS
(
P
n
)
≤
E
ABS
(
T
n
)
≤
E
ABS
(
K
1
,
n
-
1
)
with equality in the left (resp., right) inequality if and only if
T
n
is isomorphic to the path
P
n
(resp., the star
K
1
,
n
-
1
).</description><identifier>ISSN: 1598-5865</identifier><identifier>EISSN: 1865-2085</identifier><identifier>DOI: 10.1007/s12190-024-02188-z</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Apexes ; Computational Mathematics and Numerical Analysis ; Eigenvalues ; Graph theory ; Isomers ; Mathematical and Computational Engineering ; Mathematics ; Mathematics and Statistics ; Mathematics of Computing ; Original Research ; Theory of Computation ; Trees (mathematics)</subject><ispartof>Journal of applied mathematics & computing, 2024-12, Vol.70 (6), p.5483-5495</ispartof><rights>The Author(s) under exclusive licence to Korean Society for Informatics and Computational Applied Mathematics 2024. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c200t-86dc4da5a4a309779cd9a0f8c9ae30da4a44b16910f59fcb67c1003a1fe12eb03</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s12190-024-02188-z$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s12190-024-02188-z$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,776,780,27901,27902,41464,42533,51294</link.rule.ids></links><search><creatorcontrib>Lin, Zhen</creatorcontrib><creatorcontrib>Zhou, Ting</creatorcontrib><creatorcontrib>Liu, Yingke</creatorcontrib><title>On ABS Estrada index of trees</title><title>Journal of applied mathematics & computing</title><addtitle>J. Appl. Math. Comput</addtitle><description>Let
G
be a graph with
n
vertices, and
d
i
be the degree of its
i
-th vertex. The
ABS
matrix of
G
is the square matrix of order
n
whose (
i
,
j
)-entry is equal to
(
d
i
+
d
j
-
2
)
/
(
d
i
+
d
j
)
if the
i
-th vertex and the
j
-th vertex of
G
are adjacent, and 0 otherwise. Let
ρ
1
≥
ρ
2
≥
⋯
≥
ρ
n
be the eigenvalues of the
ABS
matrix of
G
. Then the
ABS
Estrada index of
G
, denoted by
E
ABS
(
G
)
, is defined as
E
ABS
(
G
)
=
∑
i
=
1
n
e
ρ
i
. In this paper, the chemical importance of the
ABS
Estrada index is investigated and it is shown that the predictive ability of
ABS
Estrada index is stronger than other connectivity Estrada indices (including Randić Estrada index, harmonic Estrada index and
ABC
Estrada index) and
ABS
index for octane isomers. Since chemical graphs of octane isomers are trees, we study the extremal problem of
ABS
Estrada index of trees, and prove that for any tree
T
n
with
n
≥
3
vertices,
E
ABS
(
P
n
)
≤
E
ABS
(
T
n
)
≤
E
ABS
(
K
1
,
n
-
1
)
with equality in the left (resp., right) inequality if and only if
T
n
is isomorphic to the path
P
n
(resp., the star
K
1
,
n
-
1
).</description><subject>Apexes</subject><subject>Computational Mathematics and Numerical Analysis</subject><subject>Eigenvalues</subject><subject>Graph theory</subject><subject>Isomers</subject><subject>Mathematical and Computational Engineering</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Mathematics of Computing</subject><subject>Original Research</subject><subject>Theory of Computation</subject><subject>Trees (mathematics)</subject><issn>1598-5865</issn><issn>1865-2085</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><recordid>eNp9kFtLAzEQhYMoWKt_QBAWfI7O5LKbPNZSL1Dog_ocsrlIi-7WZAvaX290Bd98GGY4nDMzfIScI1whQHOdkaEGCkyUQqXo_oBMUNWSMlDysMxSKyqLcExOct4A1I0GPSEXq66a3TxWizwk62217nz4qPpYDSmEfEqOon3N4ey3T8nz7eJpfk-Xq7uH-WxJHQMYqKq9E95KKywH3TTaeW0hKqdt4OCLLESLtUaIUkfX1o0rX3OLMSALLfApuRz3blP_vgt5MJt-l7py0nDkgrOaM15cbHS51OecQjTbtH6z6dMgmG8MZsRgCgbzg8HsS4iPoVzM3UtIf6v_SX0Bn8peDg</recordid><startdate>20241201</startdate><enddate>20241201</enddate><creator>Lin, Zhen</creator><creator>Zhou, Ting</creator><creator>Liu, Yingke</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>7TB</scope><scope>8FD</scope><scope>FR3</scope><scope>JQ2</scope><scope>KR7</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope></search><sort><creationdate>20241201</creationdate><title>On ABS Estrada index of trees</title><author>Lin, Zhen ; Zhou, Ting ; Liu, Yingke</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c200t-86dc4da5a4a309779cd9a0f8c9ae30da4a44b16910f59fcb67c1003a1fe12eb03</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Apexes</topic><topic>Computational Mathematics and Numerical Analysis</topic><topic>Eigenvalues</topic><topic>Graph theory</topic><topic>Isomers</topic><topic>Mathematical and Computational Engineering</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Mathematics of Computing</topic><topic>Original Research</topic><topic>Theory of Computation</topic><topic>Trees (mathematics)</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Lin, Zhen</creatorcontrib><creatorcontrib>Zhou, Ting</creatorcontrib><creatorcontrib>Liu, Yingke</creatorcontrib><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>Technology Research Database</collection><collection>Engineering Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Civil Engineering Abstracts</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Journal of applied mathematics & computing</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Lin, Zhen</au><au>Zhou, Ting</au><au>Liu, Yingke</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On ABS Estrada index of trees</atitle><jtitle>Journal of applied mathematics & computing</jtitle><stitle>J. Appl. Math. Comput</stitle><date>2024-12-01</date><risdate>2024</risdate><volume>70</volume><issue>6</issue><spage>5483</spage><epage>5495</epage><pages>5483-5495</pages><issn>1598-5865</issn><eissn>1865-2085</eissn><abstract>Let
G
be a graph with
n
vertices, and
d
i
be the degree of its
i
-th vertex. The
ABS
matrix of
G
is the square matrix of order
n
whose (
i
,
j
)-entry is equal to
(
d
i
+
d
j
-
2
)
/
(
d
i
+
d
j
)
if the
i
-th vertex and the
j
-th vertex of
G
are adjacent, and 0 otherwise. Let
ρ
1
≥
ρ
2
≥
⋯
≥
ρ
n
be the eigenvalues of the
ABS
matrix of
G
. Then the
ABS
Estrada index of
G
, denoted by
E
ABS
(
G
)
, is defined as
E
ABS
(
G
)
=
∑
i
=
1
n
e
ρ
i
. In this paper, the chemical importance of the
ABS
Estrada index is investigated and it is shown that the predictive ability of
ABS
Estrada index is stronger than other connectivity Estrada indices (including Randić Estrada index, harmonic Estrada index and
ABC
Estrada index) and
ABS
index for octane isomers. Since chemical graphs of octane isomers are trees, we study the extremal problem of
ABS
Estrada index of trees, and prove that for any tree
T
n
with
n
≥
3
vertices,
E
ABS
(
P
n
)
≤
E
ABS
(
T
n
)
≤
E
ABS
(
K
1
,
n
-
1
)
with equality in the left (resp., right) inequality if and only if
T
n
is isomorphic to the path
P
n
(resp., the star
K
1
,
n
-
1
).</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s12190-024-02188-z</doi><tpages>13</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 1598-5865 |
| ispartof | Journal of applied mathematics & computing, 2024-12, Vol.70 (6), p.5483-5495 |
| issn | 1598-5865 1865-2085 |
| language | eng |
| recordid | cdi_proquest_journals_3134326323 |
| source | Springer Nature - Complete Springer Journals |
| subjects | Apexes Computational Mathematics and Numerical Analysis Eigenvalues Graph theory Isomers Mathematical and Computational Engineering Mathematics Mathematics and Statistics Mathematics of Computing Original Research Theory of Computation Trees (mathematics) |
| title | On ABS Estrada index of trees |
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